I can't change the date inserted with Handy Timestamp. What's the reason?

Handy Timestamp creates the date and time record that can't be changed by any user. This is done on purpose. Having this information protected from changes on Confluence pages helps users to track important actions with confidence.

Can I use a hotkey to insert Handy Timestamp?

Using the key combination, you can quickly add the Handy Timestamp macro to the page. Press Ctrl+Alt+T combination when editing the page.

I don't want time to be displayed on my Handy Timestamp. I need to see the date only.

You can change the Handy Timestamp format to suit your needs. The format can be selected out of the predefined ones or entered manually.

Handy Timestamp in the Page Properties Report doesn't have the localization in my instance.

Incorrect date format for Date lozenges/@currentDate variable can be caused by the Confluence bug. You can find out more in the public Atlassian ticket CONFSERVER-58977.